Walking wrap up, 2025

December 2024

My original walking goal this year was 2,809 miles. This was to honor the perfect square 2025 = 45². 2,809 is the only square number (= 53²) between 2,800 and 2,900--my arbitrary goal range for the past few years--and it seemed like a manageable number after last year's excessive 2,865.

Too manageable. In autumn 2024, my doctor recommended that I work with a personal trainer because my knees had started making disconcerting crunching sounds when I climbed stairs. Crunch, crunch, crunch. Last December, I started working with a trainer at Empower Fitness, which is a 5.5-mile walk from home. In theory I was supposed to be strengthening muscles other than the ones I use excessively for walking, but of course I walked there and back twice a week almost every week that I was in town, and those miles added up. My knees still crunch when I climb stairs, but I've gone from barely being able to lift a 50-lb box of clay without hurting my back to deadlifting 120 pounds, so that's an improvement.

By September 2025, I thought I was more likely to land on 2,850, the triangular number I had anticipated in last year's wrap-up as a follow-up to 2024's sphenic number. 

T_n = n*(n+1)/2 
T₇₅ = 75*(75+1)/2 = 2850
Image courtesy of Melchoir in Wikipedia

Still, that seemed like blowing the only opportunity in my lifetime to walk a square number of miles in a square year. The next time a year will be square is 2116, and I don't expect to live to be 149 years old. I can walk 2,850 miles any ol' year.

I upped my goal to 2,916 miles, and enjoyed the not-quite-palindromic, reversible-numbers ring of walking 54² miles in the year 45².

I had to jack up my miles in mid-October to reach my new goal (and to bank miles for the Durham Pottery Tour), and I over-jacked. Then we spent five days on Ocracoke over Thanksgiving, where I noticed that if I jacked up my miles even more, I could make November a rare 300+ mile month. This left me a mere 155 miles short of 2,916 by the first day of December.

I confidently walked the numeral on Dec. 9--then reached it in miles on Dec. 23

Once in the habit, it's hard to cut back. I imagined my problem-solving friend N's voice in my head saying, "your goal is 2,916 recorded miles. Stop recording when you reach it." When I told her in person, she agreed her imaginary voice gave good advice. Did I heed it? Of course not.

Instead, I upped my goal to 2,965 recorded miles, a number that's nomenclaturally relevant because it's a centered square. Centered squares are figurate numbers (numbers that can be represented by dots laid out in geometric patterns) that have a single dot in the center, surrounded by dots laid out in successive outlines of squares: 

25 dots, one center dot surrounded by three successive square outlines. Image by Stefan Friedrich Birkner

13 dots demonstrating a nifty property of centered square numbers. Image by Stefan Friedrich Birkner

As the second dots-image above shows, centered squares can be expressed as sums of two consecutive perfect squares: 13 = 3² + 2². In the image above that, 25 = 4² + 3²; in my updated aspirational miles for this year, 2,965 = 39² + 38². In contrast to perfect squares, which are relevant to the mathematics of assorted disciplines, centered square numbers are of interest mainly for recreational mathematics.

While I was exploring finding a mileage higher than 2,916 that still relates to squares, I noticed that 2,965 is 49 miles more than 2,916; that is, 2,965 = 54² + 7². This made me wonder whether all positive integers can be expressed as sums of some small quantity of perfect squares (vs. e.g. tritely summing 1² until you reach the number you seek). I asked the interwebs, and lo: every positive integer can indeed be expressed as the sum of at most four perfect squares, a result known as Lagrange's Four-Square Theorem.

For the special case of a positive integer being the sum of exactly two perfect squares, as is the case for 2,965, see Fermat's theorem on the sum of two squares and this fun Sum of Two Squares calculator. Turns out 2,965 is the sum of only two combinations of two perfect squares: 54² + 7² and 39² + 38².

After learning all that, I overshot 2,965 too, because we decided to go back to Ocracoke with friends during the last week of December, and the very best thing for a walkaholic to do on Ocracoke is to scour the entire length of the beach multiple times looking for scotch bonnets. I blew past 2,965 on Dec. 28, with three entire days of walking left to go--but thanks to the four-square theorem, I figured any integer's-worth of miles could be a celebration of squares.

Mission accomplished

Whereas last year I came precipitously close to violating my 2023 goal to not be quite so obsessive about walking, this year I just went ahead and violated it. 

Final totals for 2025: 10 scotch bonnets (3 over Thanksgiving, 7 [!] in December) and 55² recorded miles, a new record for me. And if it so happened that the siren song of the scotch bonnet and the annual Ocracoke bird count called to me on this sunny last day of 2025--well then, I am fortunate to have wise friends like N.*

Next year's goal: 2026 is a "semiprime," a.k.a. "biprime." Semiprime numbers have only two factors, excluding 1 and the number itself, e.g. 2,026 = 2 * 1,013. (Incidentally, 2,809 = 53 * 53 is also semiprime--the prime factors don't need to be distinct.) Walking a semiprime number of miles has been on my to-do list as a follow-up to walking a sphenic number, and there are several options between 2,800 and 2,900.

And the year after that? 2027 is prime.

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2025 walking highlights:

Longest single walk: 31.22 miles. This was my annual American Tobacco Trail long walk on January 18. I mentioned to a friend that two years ago I had extended the 24.5-mile route to 26.2 so I could say I had walked a marathon, and that last year I had brought that up to 28. She noted that three more miles would be 50k. So I walked my first and probably only 50k ever. Turns out that once you hit ~26 miles using a 24.5-mile path and then some, there's not much entertainment value in going five miles farther. I don't know how ultramarathoners do it.

Not doing this again

Number of weeks topping 100 miles: 1 (early June).

Most miles in a single Monday-Sunday week: 101, thanks to a combination of hiking the Malerweg and intentionally walking some extra miles in Berlin to reach 100+.

Number of months topping 300 miles: 3 (June, July, and November). Haven't managed that since 2022. Hooray for long summer hikes and November beaches.

Most miles in a single month: 313.8 (November).

Favorite multi-day walk: Rorschach to Gündlischwand in 15 days.

Favorite single-day walk: This is a new category this year, so I don't have to choose between the Malerweg and Switzerland for the multi-day walk. On June 2, S and I hiked up Kaiserkrone in Saxony and gazed across Nebelmeer 217 years into the past.

Afterward, S caught a train back to Dresden, and I hiked the section of the Malerweg between Rathen and Liebethaler Grund. Bestill my beating heart...

Caspar David Friedrich was here.

Favorite photo: see "favorite single-day walk" above. Runners-up:

Obersee at dawn on day 6 of our hike from Rorschach to Gündlischwand

Stormy day on the Wörthsee after sailing the model boat.

Nothing else new to report. All of the walking philosophy from the preceding walking-intense years still stands, as does the warning from my PCP that knee cartilage has a limited lifespan, so don't overdo it. Ha.

What does 2026 hold in store? Less walking than in 2025. Another American Tobacco Trail marathon. Another car-free February. And a 17-day hut-to-hut hike from the Dolomites to the Adriatic. 

Here's to great walking in 2026!

*Below, ChatGPT's attempt at representing the figurate number of recorded miles with which I actually ended 2025. This illustrates that (1) ChatGPT isn't great at representing figurate numbers, because going by the number of dots in the border, the star is short by ~984 dots (and some of the "dots" are less dots than dot-like), and (2) someone has zero self-restraint when it comes to walking on undeveloped beaches.

Walking wrap up, 2024

For reasons explained elsewhere, my goal in 2024 was to walk a sphenic number of recorded miles in the vicinity of my 2023 mileage. 

I closed last year with 2,814 miles. This year, I aimed for 2,821 ( = 7 * 13 * 31). I figured if I overshot by a mile, I'd still end with a sphenic number (2,822 = 2 * 17 * 83). And if I overshot that by a mile, I'd land on a semiprime (2,823 = 3 * 941)--not ideal, but whatevs.¹

I was behind schedule in November, but a hiking-focused Thanksgiving trip to Tucson helped put 2,821 back in the realm of obtainable, so I stepped up my miles in December. 

I hit 2,821 on the Atlantic coast of Shackleford Banks, NC, on December 27.

Likewise 2,822...

...and 2,823.

So much for that.

I considered giving myself a new sphenic goal, or a semiprime, or a prime, and then Wikipedia told me that 2,850 is a triangular number. It isn't sphenic (although some triangular numbers are), but I figure a triangular number is in the spirit of the word sphenic--a sphenic number's etymological equilateral cousin, so to speak--so 2,850 became my new goal.²

T_n = n*(n+1)/2
T₇₅ = 2850 = 75*(75+1)/2
(Image courtesy of Melchoir on Wikipedia)

By the time I decided T₇₅ should be my goal, I had three days and 11 miles left, so of course I overshot that too.

Final recorded mileage for 2024 = 2865 miles. 2865 = 3 * 5 * 191, a sphenic number.

My last walk of 2024 was 0.2 miles longer than necessary

This is way more miles than I intended, and getting there brought me precipitously close to violating my 2023 goal to not be quite so obsessive about walking. But walking keeps me sane when SAD kicks in, so it's all good.

Next year's goal: in honor of 2025 = 45², I'll initially aim for 2,809 = 53². Overshooting options include the triangular³ number 2,850 or a semiprime.

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Some 2024 walking highlights:

Longest single walk: 28.09 miles. This was my annual American Tobacco Trail long walk on February 4, with a little extra distance to ensure it was longer than last year's ATT marathon. (I did the ATT again in mid-April, but didn't walk quite as far.) 

Number of weeks topping 100 miles: just 1 (in early June). I had 4 in 2023. I felt like I took more rest days this year than last, but apparently I did better keeping up with my weekly mileage goal (50 miles) and more forcefully made up for the lapses (18 weeks under 50 miles this year, vs. 20 last year).

Most miles in a single week: 100.3, which I only reached because I felt obligated to go for a 1.75 mile walk at 10pm on June 9. 

Number of months topping 300 miles: 2 (June and July), same as last year.

Most miles in a single month: 337.3 (June)

Favorite multi-day walk: Sterzing to Feltre in 13 days. S and I started in Sterzing, where we left off in 2023, and declared the Alps crossed when we arrived in Feltre. Feltre is described on various websites as being in the foothills of the Alps (although those hills still looked pretty mountainous to us), and is where the Alta Via 2 delle Dolomiti terminates.

Favorite photo: Mein geliebter Wanderer ueber dem Nebel(isch)meer, taken on Day 9 on the descent from Rifugio delle Selle:

There were some fine runners up:

Day 5, on Seceda

Day 4, en route to Rifugio Resciesa

Day 12, atop Mt. Pavione. The photo does justice neither to the expanse of globeflowers nor to the double cirque.

Day 8, heroic S literally moving mountains to help me cross a river of snow-melt

Nothing else new to report. All of the walking philosophy from the preceding walking-intense years still stands.

What does 2025 hold in store? Another American Tobacco Trail marathon. Another car-free February. And hopefully the Malerweg--a loop hike in the footsteps of Caspar David Friedrich (and other Romantic-era artists), to see the geology that inspired paintings like Der Wanderer ueber dem Nebelmeer--and a 2-week point-to-point trek in German-speaking Switzerland.

Here's to great walking in 2025!

My serifs weren't as good this year as last...

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¹I walked a 4-almost-prime number last year, so walking a sphenic number this year was supposed to make it obvious to aim for a semiprime in 2025 and a prime in 2026. I'm all about long-term goals.

²In case you missed it, the word "sphenic" comes from the Greek word σφήνα, meaning "wedge." 

³Of course, there are also square numbers (like 2025 and 2809), and pentagonal numbers, and hexagonal and etc. polygonal numbers, all of which are fun to think about as mileage goals, as well as polyhedral numbers that my brain has a harder time wrapping itself around.

Walking wrap up, 2023

My goal in 2023 was to walk less than in 2022, since my 2022 mileage (3,024 miles) required an obsessiveness I didn't want to repeat. 

Goal achieved! By 4pm on New Year's Eve, I had reached 2,813 recorded miles, which is indeed less than 3,024, but still enough to say "I walked a lot."

Then I thought, "but wouldn't it be cool to have a mile total that's divisible by 3? And by 2?" So at 10pm, after dinner with friends yet before sitting around a fire pit with them waiting for midnight, S and I excused ourselves to walk a tidy figure 8 around our block and an adjacent block, and I ended the year with 2,814 miles.

2,814 is indeed divisible by 3 and by 2--and by 7 and 67. Wouldja look at that--it's the product of four distinct prime numbers! What fun! I wonder how many of those are there in the vicinity of 2,800?¹ 

Had I been paying more attention, I might have happily stopped walking at 2,813 miles, since 2,813 is a semiprime--the product of just two prime numbers (2813 = 29 * 97). By definition, those prime numbers don't have to be distinct--so the squares of prime numbers are also semiprimes.

What about products of three prime numbers? A positive integer having three distinct prime factors is called a sphenic number (by definition, no squares are allowed). Once you go from three to four prime factors, you enter the territory of k-almost primes, where k is the number of prime factors. Squares are allowed, so 2,814, with its four distinct prime factors, is 4-almost prime, but so is 16 (16 = 2 * 2 * 2 * 2). I have mixed feelings about this.

I think for 2024 I should aim for my mileage total to be a sphenic number.³

I asked ChatGPT for a list of sphenic numbers between 2,800 and 3,200, and learned that ChatGPT is as incompetent at listing sphenic numbers as it is at reading roadmaps. This both gives me hope and stokes fear: AI will never replace people, but reliance on AI will probably end civilization as we know it.

To the sphenic list rescue: The On-Line Encyclopedia of Integer Sequences! Had I stopped at 2,810 miles last year, I would have nailed it.

For those of you dear readers who are still hanging in there, here's one last, exciting tidbit tangentially related to the number 2,814. I find it delightful how easy it is to see that 7 is a factor of 2,814, compared to e.g. the less immediately obviously 7-factorable number 2,226 (also a 4-almost-prime with four distinct factors). So I asked Google if there's an easy way to quickly determine whether a number is divisible by 7, AND THERE IS!² Oh my goodness, that extra figure-8 mile at 10pm on New Year's Eve was worthwhile!

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Here are the 2023 walking stats I'm most proud of:

Longest single walk: 26.81 miles. I finally walked a marathon! Twice, actually, on January 15 and January 27. The first was my annual American Tobacco Trail long walk. The second was a repeat for a friend who had wanted to go the first time.

Number of weeks topping 100 miles: 4 (2 in June, 1 in July, 1 in November)

Most miles in a single week: 108 (July)

Number of months topping 300 miles: 2 (June and July)

Most miles in a single month: 361 (June)

Favorite multi-day walk: In June, S and I went out the front door of the familial cottage in southern Bavaria and spent 11 days hiking to Italy.

Favorite photo: Hiking up from Hallerangerhaus to the Lafatscher saddle on Day 6 of that multi-day hike: 

Nothing else new to report. All of the walking philosophy from the preceding walking-intense years still stands.

What does 2024 hold in store? Another American Tobacco Trail marathon. Another car-free February. Hopefully completing our multi-day trek southward across the Alps, from our 2023 end point to where the mountains yield to flat terrain. 

Here's to great walking in 2024!

¹Between 2,800 (my minimum mileage goal last year) and 2,850, there are three almost-primes that have four distinct prime factors: 2,805 (5 * 3 * 11 * 17); 2,814 (2 * 3 * 7 * 67); and 2,838 (2 * 3 * 11 * 43).

²To quickly determine whether a number is divisible by 7: 

(1) remove the last digit of the number; 

(2) double that digit; then 

(3) subtract that from the truncated original number. 

If the result is divisible by 7, then the original number is divisible by 7. 

For example, 2,226:

(1) remove the 6;

(2) double it (6 * 2 = 12); then

(3) subtract 12 from the truncated number (222 - 12 = 210). 

210 is divisible by 7, so 2,226 is divisible by 7. 

My mathematician dad liked general proofs over anecdotal examples, so I'm including this link for him, wherein a general proof is offered.

(And about the above example, what fun: 210 = 2 * 3 * 5 * 7 -- the product of the four smallest prime numbers. 2,226 is likewise the product of four distinct primes -- 2 * 3 * 7 * 59. Clearly 2023 has given me food for thought for my 2024 walks.)

³I was interested in the etymology of the word sphenic, so I looked it up in the etymology bible, a.k.a. the Oxford English Dictionary--and there's no entry for the word at all. Tsk. Wikipedia to the rescue: the word is from Ancient Greek σφήν (sphḗn, “wedge”), + -ic.

45th anniversary of NC's Mountains-to-Sea Trail photo dump

This September is the 45th anniversary of North Carolina's Mountains-to-Sea Trail. Friends of the MST has encouraged folks to celebrate by hiking 45 MST miles this month. As I hadn't been away from Durham since July 31, I took this as motivation to hike out of town.

Part I: "to," Saturday Sept. 10

My plan was to start hiking the second weekend of September, with a section of the MST in Doughton Recreation Area, but hurricane Earl, brewing far off in the Atlantic, had other plans, drenching much of NC in rain. Yet the forecast in Durham was decent until that Saturday afternoon--so plans switched to an Eno River hike instead, from the Pleasant Green access to Penny's Bend. Friends joined me for the first 5 miles. The weather held up until the last half hour, but it was fun to get drenched knowing it would just be for a short time. Thanks to S for providing shuttle service.

Shrooms were abundant on the trail. Size 10.5W women's sneaker included for scale:

The fireplace mantel was decorated for some special occasion:

 

View from the bridge over Guess Road:

Wildflowers were abundant--some native and some not...

This spider lily isn't native to the U.S.

Thanks, blazes!

This copperhead was sheltering its head from the rain. Since it couldn't see me, obviously I couldn't see it. 

Part II: "Sea," Friday September 16-Sunday September 18

The next Friday, I headed to Ocracoke. My original plan was to drive to the north end of Ocracoke on Saturday and walk back to Ocracoke village, then walk back to the north end on Sunday to pick up the car. But I arrived late afternoon on Friday, and how can one resist a walk on the beach? I ended up covering almost 11 miles that evening, from town to Ramp 67 and back again, so I sucked it up and completed the rest of the island on Saturday morning (Pony Pen access to the north end ferry dock, back south to Ramp 67, and then back to Pony Pen). On Sunday, I treated myself to a mostly non-MST walk to the south end of the island, where I finally found the sand dollar I had been searching for for a friend.

Friday evening:

Sea turtle nests abounded:

Saturday morning:

See the sun just coming up on the horizon? 

Looking south...

My rule on the beach is that if plastic and I make eye contact, I have to pick it up. Most of the trash I encountered this time was limited to the road toward the ferry dock at the north end of the island. I'm not sure it's correct to call this a "bright side," but generally the beach provides clean, empty plastic bags along with the rest of the trash, so after I deposited my heading-north trash in the restroom garbage can, the beach said "hey, here's another bag for the return trip."

One less straw to get stuck in a sea turtle's schnozz, and one
less plastic bag to be mistaken for a jellyfish, at least for now. 

Hope springs eternal

I started my walk north at low tide. I finished the last stretch--south from Pony Pen to Ramp 67 and back again--near high tide, which meant less traction and steeper sloping beach. It. Was. A. Slog. But that slog meant I had covered almost the entire length of the island twice in two days.  

All that was left was the southern end on...

Sunday morning (mostly non-MST):

Here's the only whole sun-bleached sand dollar I found the entire weekend: 

South end of Ocracoke: ocean to the left, sound to the right:

I found a few whole, fuzzy, unbleached sand dollars too:

If you can't tell by now, I'm a fan of reflections:

This shell was still occupied:

Sunday afternoon (back to MST):

The beach colors on Hatteras were noticeably different from the beach colors on Ocracoke, at least for the mile south of Ramp 27, where I decided to take a break from driving.

Near Ramp 27, I encountered the remains of the George A. Kohler, built in 1919, wrecked in 1933. A history of the ship is here.

More orange-browns than on Ocracoke

Part III: "Mountains," Saturday September 24

By the end of my Ocracoke trip, I had reached the 45-mile goal and then some, but had only covered the "Sea" and "to" parts of the trail (poor Eno River, reduced to a preposition!). So yesterday, S and I drove west to Doughton Recreation Area for a 12.5 mile "Mountains" out-and-back hike. The forecast was "partly sunny" with a 10% chance of rain; thus we learned about mountain micro-climates, and were grateful we had both brought rain jackets with us.

Of all the stretches of MST I hiked this month, the stretch in Doughton Recreation Area was by far the best marked:

Given enough time, trees will eat anything:

Autumn! Lots of yellow leaves on the ground, but not many red ones yet...

Brinegar cabin (built 1880) interpretive exhibit:

When I took this photo, I was seeing the ladybug...

...not the aphids:

Best blaze ever, yip yip yipyipypyip:

Doughton gives Bavaria a run for its lone-tree-on-a-hill money:

Carly came with us, of course: